Kinematical Algebras Research Articles

GENERALIZED QUANTUM RELATIVISTIC KINEMATICS: A STABILITY POINT OF VIEW

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincaré-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

Schrödinger invariance and spacetime symmetries

The free Schrödinger equation with mass M can be turned into a non-massive Klein–Gordon equation via Fourier transformation with respect to M . The kinematic symmetry algebra sch d of the free d-dimensional Schrödinger equation with M fixed appears therefore naturally as a parabolic subalgebra of the complexified conformal algebra conf d+2 in d+2 dimensions. The explicit classification of the parabolic subalgebras of conf 3 yields physically interesting dynamic symmetry algebras. This allows us to propose a new dynamic symmetry group relevant for the description of ageing far from thermal equilibrium, with a dynamical exponent z=2. The Ward identities resulting from the invariance under conf d+2 and its parabolic subalgebras are derived and the corresponding free-field energy–momentum tensor is constructed. We also derive the scaling form and the causality conditions for the two- and three-point functions and their relationship with response functions in the context of Martin–Siggia–Rose theory.

Representations of quantum bicrossproduct algebras

We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1+1)--dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum kappa Galilei algebra.

Induced representations of quantum kinematical algebras and quantum mechanics

Unitary representations of kinematical symmetry groups of quantum systems are fundamental in quantum theory. We propose in this paper its generalization to quantum kinematical groups. Using the method, proposed by us in a recent paper (olmo01), to induce representations of quantum bicrossproduct algebras we construct the representations of the family of standard quantum inhomogeneous algebras $U_\lambda(iso_{\omega}(2))$. This family contains the quantum Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As byproducts we obtain the actions of these quantum algebras on regular co-spaces that are an algebraic generalization of the homogeneous spaces and $q$--Casimir equations which play the role of $q$--Schr\"odinger equations.

Induced representations of quantum kinematical algebras

We construct the induced representations of the null-plane quantum Poincar\'e and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure makes use of the concept of module and is based on the existence of a pair of Hopf algebras with a nondegenerate pairing and dual bases.

Graded contractions of Lie algebras and central extensions*

We use the concept of grading of Lie algebras to investigate the appearance of central charges during the contraction process. As for the usual graded contractions, one finds simultaneously the central extensions of classes of algebras, rather than specific Lie algebras. To illustrate the method, we consider in detail two physical applications: the kinematical algebras of spacetime and the u(n)-bosons limits of some Lie algebras.

2 + 1 kinematical expansions: from Galilei to de Sitter algebras

Expansions of Lie algebras are the opposite process of contractions. Starting from a Lie algebra, the expansion process goes to another one, non-isomorphic and less abelian. We propose an expansion method based in the Casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces. This method is applied for expansions within the family of Lie algebras of 3d spaces and (2+1)d kinematical algebras. We show that these expansions are classed in two types. The first type makes different from zero the curvature of space or space-time (i.e., it introduces a space or universe radius), while the other has a similar interpretation for the curvature of the space of worldlines, which is non-positive and equal to $-1/c^2$ in the kinematical algebras. We get expansions which go from Galilei to either Newton--Hooke or Poincar\'e algebras, and from these ones to de Sitter algebras, as well as some other examples.

Kinematics and uncertainty relations of a quantum test particle in a curved space-time

A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V_e of a curved space-time can be equipped with a nonassociative binary operation called the geodesic multiplication of space-time points. In the case of the Minkowski space-time, left and right translations of the geodesic multiplication coincide and amount to a rigid shift of the space-time x->x+a. In a curved space-time infinitesimal geodesic right translations can be used to define the (geodesic) momentum operators. The commutation relations of position and momentum operators are taken as the quantum kinematic algebra. As an example, detailed calculations are performed for the space-time of a weak plane gravitational wave. The uncertainty relations following from the commutation rules are derived and their physical meaning is discussed.

Quantization of kinematics and dynamics on with difference operators and a relatedq-deformation of the Witt algebra

Motivated by the fact that momentum observables which are modelled in a classical theory as difference quotients of functions are directly accessible to measurements, whereas differentials are mathematical idealizations, which are obtained from difference quotients via a limiting process, we propose a quantization method in which momentum operators are given in terms of q-derivatives, i.e. a particular type of difference quotient, which is particularly suitable on . The quantization scheme is obtained from Borel quantization, in which momentum operators are given as differential operators, via a q-deformation of the kinematical algebra. It will be applied to a system localized and moving on the N-point discretization of and leads to a discrete, nonlinear Schrodinger equation. In the limit , i.e. the continuous idealization, we find evolution equations which are special cases of the nonlinear Schrodinger equation derived from Borel quantization on , which is based on the undeformed kinematical algebra. It turns out that with this procedure both the real and imaginary part of the nonlinearity can be derived, which without deformation was only possible for the imaginary part. Hence, one can learn on the situation in the continuous case if it is viewed as a limit of the (q-deformed) case.

Quantum kinematics of a test particle in a curved spacetime

A possible model for the quantum kinematics of a test particle in curved spacetime is proposed. Every reasonable neighbourhood of a curved spacetime can be equipped with a non-associative binary operation called geodesic multiplication. Its infinitesimal right translations are used to define the (geodesic) momentum operators. The corresponding commutation relations are taken as the quantum kinematic algebra. It coincides with the usual canonical Poisson algebra (Weyl's kinematics) only in the case of flat spacetime. A BRST-like operator is constructed and its physical meaning is discussed. As an example, detailed calculations are performed for the spacetime of a weak plane gravitational wave.

Casimir invariants for the complete family of quasisimple orthogonal algebras

A complete choice of generators of the centre of the enveloping algebras of real quasisimple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras , as well as the Casimirs for many non-simple algebras such as the inhomogeneous , the Newton - Hooke and Galilei type, etc, which are obtained by contraction(s) starting from the simple algebras . The dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction. The structure of the higher-order invariants is given in a convenient `pyramidal' manner, in terms of certain sets of `Pauli - Lubanski' elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3 + 1)-kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.

The general solution of the real graded contractions of

The general solution of the graded contraction equations for a grading of the real compact simple Lie algebra so(N+1) is presented in an explicit way. It turns out to depend on independent real parameters. The structure of the general graded contractions is displayed for the low-dimensional cases, and kinematical algebras are shown to appear straightforwardly. The geometrical (or physical) meaning of the contraction parameters as curvatures is also analysed; in particular, for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous spacetimes.

Graded contractions of complex simple lie algebraB 2 and its representations

I describe the toroidal contractions of the complex simple Lie algebraB 2 and its representations, by using the theory of elements of finite order (EFO) of compact Lie groups to obtain the gradings. The non-toroidal Z2 ⊗ Z2 ⊗ Z2 ⊗Z 2 finest grading is also given, because its contractions include the Poincare and Galilei algebras, and other kinematical algebras.

Non-standard quantum so(2,2) and beyond

A new "non-standard" quantization of the universal enveloping algebra of the split (natural) real form $so(2,2)$ of $D_2$ is presented. Some (classical) graded contractions of $so(2,2)$ associated to a $Z_2 \times Z_2$ grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of "non-standard" quantum algebras. Some of these algebras can be realized as (2+1) kinematical algebras; we explicitly introduce a new deformation of Poincar\'e algebra, which is naturally linked to the null plane basis. Another realization of these quantum algebras as deformations of the conformal algebras for the two-dimensional Euclidean, Galilei and Minkowski spaces is given, and its new properties are emphasized.

Quantum (2+1) kinematical algebras: a global approach

In this paper we give an approach to quantum deformations of the (2+1) kinematical Lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in N=3 dimensions. We cover at once all the kinematical geometries including the quantum versions of Inonu-Wigner contractions, which are defined in a natural way and relate q-deformations as expected. We thus obtain some q-deformations previously known for the three-dimensional Euclidean and (2+1)-Poincare algebras and also some new q-deformations for these and other kinematical algebras, such as the (2+1)-de Sitter, Galilei and Newton-Hooke algebras.

The classical limit of so(3) compact quantum systems

In a paper by Fawcett and Bracken (1991), the classical limit of ordinary non-compact quantum systems is considered in terms of the contraction of the underlying kinematical Lie algebra (the Weyl-Heisenberg algebra) and its representations to the Abelian Lie algebra of the same dimension and its representations. Some of those ideas are adapted to discuss in similar terms, the classical limit of compact quantum systems whose underlying kinematical algebras are of the special orthogonal type. The classical dynamical system that results from the classical limit of compact quantum mechanical system will be called a 'compact classical system'. Poisson brackets for such 'compact classical systems' have already been given in the literature and the recovery of the Poisson bracket for so(3) compact classical systems is demonstrated in terms of the contraction limit.

The classical limit of quantum mechanics as a Lie algebra contraction

The classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non-compact kinematical algebra w1 (the Weyl-Heisenberg algebra) to the three-dimensional Abelian algebra t3. An appropriate definition of the contraction of a Lie algebra and of a sequence of its representations is given. For some quantum systems with simple explicitly integrable dynamics, it is shown how the classical Poisson bracket and classical trajectories are obtained in the limit. Each trajectory is associated with a one-dimensional representation of t3 within a direct integral of such representations in a Hilbert space. Each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics, and a superselection rule arises naturally between any two such subspaces.

Galilean quantum kinematics

The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.

Quantum kinematics of the harmonic oscillator

The formalism of non-Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space-time realization of the group, casts the Schrödinger operator into the familiar time-dependent space-time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time-dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space-time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.

Compact quantum systems: Internal geometry of relativistic systems

A generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Qi and corresponding momentum operators Pi, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.